A turbulent plume from a continuous source of buoyancy in a long tank is shown to generate a series of quasi-steady counterflowing horizontal shear layers throughout the tank. Both the horizontal flow velocity and the depth of the shear layers are observed to decrease with distance above/below the plume outflow. The shear layers are supported by the stable density stratification produced by the plume and are superimposed on the vertical advection and entrainment inflow that make up the so-called ‘filling box’ circulation. Thus, at some depths, the surrounding water flows away from the plume instead of being entrained, although we see no evidence of ‘detrainment’ of dense plume water. Given the stratification produced by the plume at large times, the timescale for the velocity structure to adjust to changes in forcing is proportional to the time for long internal gravity waves to travel the length of the tank. The shear layers are interpreted in terms of internal normal modes that are excited by, and which in turn determine, the horizontal plume outflow. The sixth and seventh baroclinic modes typically dominate because at the level of the plume outflow their phase speed is approximately equal and opposite to the vertical advection in the ‘filling box’. Also, the approximate balance between phase speed and advection is found to hold throughout the tank, resulting in the observed quasi-steady flow structure. Viscosity causes the horizontal velocity in the shear layers to decrease with distance above/below the plume outflow, and is thought to be responsible for a low-frequency oscillation in the flow structure that is observed during experiments.